Toniolo and Linder, Equation (13)

Percentage Accurate: 49.2% → 62.0%

Time: 24.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 49.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 62.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{{\ell}^{2}}{Om}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_4 := t_3 \cdot \left(U - U*\right)\\ t_5 := \sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_3 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_5 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\mathsf{fma}\left(t_1, -2, t\right) - t_4\right)}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - t_4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot \left(t + t_1 \cdot -2\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (/ (pow l 2.0) Om))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* n (pow (/ l Om) 2.0)))
        (t_4 (* t_3 (- U U*)))
        (t_5 (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) (* t_3 (- U* U)))))))
   (if (<= t_5 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U (- (fma t_1 -2.0 t) t_4))))
     (if (<= t_5 INFINITY)
       (sqrt (* t_2 (- (- t (* 2.0 (* l (/ l Om)))) t_4)))
       (sqrt (* 2.0 (fabs (* U (* n (+ t (* t_1 -2.0)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = pow(l, 2.0) / Om;
	double t_2 = (2.0 * n) * U;
	double t_3 = n * pow((l / Om), 2.0);
	double t_4 = t_3 * (U - U_42_);
	double t_5 = sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + (t_3 * (U_42_ - U)))));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * (fma(t_1, -2.0, t) - t_4)));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) - t_4)));
	} else {
		tmp = sqrt((2.0 * fabs((U * (n * (t + (t_1 * -2.0)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64((l ^ 2.0) / Om)
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_4 = Float64(t_3 * Float64(U - U_42_))
	t_5 = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_3 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * Float64(fma(t_1, -2.0, t) - t_4))));
	elseif (t_5 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) - t_4)));
	else
		tmp = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * Float64(t + Float64(t_1 * -2.0)))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * N[(N[(t$95$1 * -2.0 + t), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * N[(t + N[(t$95$1 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 0.0

   1. Initial program 7.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 7.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   5. Applied egg-rr 41.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 41.0%

         \[\leadsto \sqrt{\color{blue}{n \cdot 2}} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)} \]

      2. fma-def 41.0%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)} \]

      3. associate--r+ 41.0%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}} \]

      4. cancel-sign-sub-inv 41.0%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(\color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \]

      5. metadata-eval 41.0%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \]

      6. +-commutative 41.0%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \]

      7. *-commutative 41.0%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2} + t\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \]

      8. fma-def 41.0%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)} \]

      9. associate-*r* 44.4%

         \[\leadsto \sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)} \]

   7. Simplified 44.4%

      \[\leadsto \color{blue}{\sqrt{n \cdot 2} \cdot \sqrt{U \cdot \left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 65.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 68.3%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 68.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 3.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 3.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}\right)}} \]

      2. sqrt-unprod 0.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{\left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}}} \]

      3. pow2 0.5%

         \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{{\left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}^{2}}}} \]

   5. Applied egg-rr 1.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{{\left(n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{2}}}} \]
   6. Step-by-step derivation

      1. unpow2 1.4%

         \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{\left(n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \cdot \left(n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}}} \]

      2. rem-sqrt-square 1.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left|n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right|}} \]

      3. fma-def 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)\right)\right|} \]

      4. associate--r+ 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)\right|} \]

      5. cancel-sign-sub-inv 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      6. metadata-eval 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      7. +-commutative 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      8. *-commutative 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2} + t\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      9. fma-def 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      10. associate-*r* 1.4%

          \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right|} \]

   7. Simplified 1.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left|n \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right|}} \]

   8. Taylor expanded in n around 0 39.7%

      \[\leadsto \sqrt{2 \cdot \left|\color{blue}{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right|}\\ \end{array} \]

Alternative 2: 60.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \sqrt{t_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_2 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - t_2 \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (* n (pow (/ l Om) 2.0)))
        (t_3 (sqrt (* t_1 (+ (- t (* 2.0 (/ (* l l) Om))) (* t_2 (- U* U)))))))
   (if (<= t_3 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U t)))
     (if (<= t_3 INFINITY)
       (sqrt (* t_1 (- (- t (* 2.0 (* l (/ l Om)))) (* t_2 (- U U*)))))
       (sqrt (* 2.0 (fabs (* U (* n (+ t (* (/ (pow l 2.0) Om) -2.0)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = n * pow((l / Om), 2.0);
	double t_3 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) + (t_2 * (U_42_ - U)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) - (t_2 * (U - U_42_)))));
	} else {
		tmp = sqrt((2.0 * fabs((U * (n * (t + ((pow(l, 2.0) / Om) * -2.0)))))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = n * Math.pow((l / Om), 2.0);
	double t_3 = Math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) + (t_2 * (U_42_ - U)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) - (t_2 * (U - U_42_)))));
	} else {
		tmp = Math.sqrt((2.0 * Math.abs((U * (n * (t + ((Math.pow(l, 2.0) / Om) * -2.0)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = (2.0 * n) * U
	t_2 = n * math.pow((l / Om), 2.0)
	t_3 = math.sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) + (t_2 * (U_42_ - U)))))
	tmp = 0
	if t_3 <= 0.0:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	elif t_3 <= math.inf:
		tmp = math.sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) - (t_2 * (U - U_42_)))))
	else:
		tmp = math.sqrt((2.0 * math.fabs((U * (n * (t + ((math.pow(l, 2.0) / Om) * -2.0)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_3 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_2 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) - Float64(t_2 * Float64(U - U_42_)))));
	else
		tmp = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * Float64(t + Float64(Float64((l ^ 2.0) / Om) * -2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = (2.0 * n) * U;
	t_2 = n * ((l / Om) ^ 2.0);
	t_3 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) + (t_2 * (U_42_ - U)))));
	tmp = 0.0;
	if (t_3 <= 0.0)
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	elseif (t_3 <= Inf)
		tmp = sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) - (t_2 * (U - U_42_)))));
	else
		tmp = sqrt((2.0 * abs((U * (n * (t + (((l ^ 2.0) / Om) * -2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * N[(t + N[(N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 0.0

   1. Initial program 7.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 7.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 7.2%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]

   5. Taylor expanded in t around inf 23.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 7.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. *-commutative 7.2%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

      3. associate-*l* 23.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   7. Simplified 23.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 23.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}} \]

      2. sqrt-prod 34.8%

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}} \]

      3. *-commutative 34.8%

         \[\leadsto \sqrt{\color{blue}{n \cdot 2}} \cdot \sqrt{U \cdot t} \]

   9. Applied egg-rr 34.8%

      \[\leadsto \color{blue}{\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 65.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 68.3%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 68.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 3.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 3.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}\right)}} \]

      2. sqrt-unprod 0.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{\left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}}} \]

      3. pow2 0.5%

         \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{{\left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}^{2}}}} \]

   5. Applied egg-rr 1.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{{\left(n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{2}}}} \]
   6. Step-by-step derivation

      1. unpow2 1.4%

         \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{\left(n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \cdot \left(n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}}} \]

      2. rem-sqrt-square 1.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left|n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right|}} \]

      3. fma-def 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)\right)\right|} \]

      4. associate--r+ 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)\right|} \]

      5. cancel-sign-sub-inv 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      6. metadata-eval 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      7. +-commutative 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      8. *-commutative 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2} + t\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      9. fma-def 1.4%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      10. associate-*r* 1.4%

          \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right|} \]

   7. Simplified 1.4%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left|n \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right|}} \]

   8. Taylor expanded in n around 0 39.7%

      \[\leadsto \sqrt{2 \cdot \left|\color{blue}{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right|}\\ \end{array} \]

Alternative 3: 60.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_3 := \sqrt{t_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_2 \cdot \left(U* - U\right)\right)}\\ \mathbf{if}\;t_3 \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_1 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - t_2 \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2 (* n (pow (/ l Om) 2.0)))
        (t_3 (sqrt (* t_1 (+ (- t (* 2.0 (/ (* l l) Om))) (* t_2 (- U* U)))))))
   (if (<= t_3 0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U t)))
     (if (<= t_3 INFINITY)
       (sqrt (* t_1 (- (- t (* 2.0 (* l (/ l Om)))) (* t_2 (- U U*)))))
       (pow (* (fma (/ (pow l 2.0) Om) -2.0 t) (* 2.0 (* n U))) 0.5)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = n * pow((l / Om), 2.0);
	double t_3 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) + (t_2 * (U_42_ - U)))));
	double tmp;
	if (t_3 <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * ((t - (2.0 * (l * (l / Om)))) - (t_2 * (U - U_42_)))));
	} else {
		tmp = pow((fma((pow(l, 2.0) / Om), -2.0, t) * (2.0 * (n * U))), 0.5);
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_3 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_2 * Float64(U_42_ - U)))))
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) - Float64(t_2 * Float64(U - U_42_)))));
	else
		tmp = Float64(fma(Float64((l ^ 2.0) / Om), -2.0, t) * Float64(2.0 * Float64(n * U))) ^ 0.5;
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0 + t), $MachinePrecision] * N[(2.0 * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 0.0

   1. Initial program 7.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 7.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 7.2%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]

   5. Taylor expanded in t around inf 23.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 7.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. *-commutative 7.2%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

      3. associate-*l* 23.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   7. Simplified 23.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 23.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}} \]

      2. sqrt-prod 34.8%

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}} \]

      3. *-commutative 34.8%

         \[\leadsto \sqrt{\color{blue}{n \cdot 2}} \cdot \sqrt{U \cdot t} \]

   9. Applied egg-rr 34.8%

      \[\leadsto \color{blue}{\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < +inf.0

   1. Initial program 65.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 68.3%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 68.3%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 3.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 1.0%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]
   5. Step-by-step derivation

      1. pow1/2 35.6%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 35.6%

         \[\leadsto {\color{blue}{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}}^{0.5} \]

      3. cancel-sign-sub-inv 35.6%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)}\right)}^{0.5} \]

      4. metadata-eval 35.6%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right)\right)}^{0.5} \]

      5. unpow2 35.6%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)}^{0.5} \]

      6. associate-*l/ 38.5%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right)\right)}^{0.5} \]

      7. associate-/r/ 38.5%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(t + -2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right)\right)}^{0.5} \]

      8. +-commutative 38.5%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\left(-2 \cdot \frac{\ell}{\frac{Om}{\ell}} + t\right)}\right)}^{0.5} \]

      9. associate-/r/ 38.5%

         \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(-2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)} + t\right)\right)}^{0.5} \]

      10. associate-*l/ 35.6%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(-2 \cdot \color{blue}{\frac{\ell \cdot \ell}{Om}} + t\right)\right)}^{0.5} \]

      11. unpow2 35.6%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(-2 \cdot \frac{\color{blue}{{\ell}^{2}}}{Om} + t\right)\right)}^{0.5} \]

      12. *-commutative 35.6%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2} + t\right)\right)}^{0.5} \]

      13. fma-udef 35.6%

          \[\leadsto {\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)}\right)}^{0.5} \]

   6. Applied egg-rr 35.6%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot \left(n \cdot U\right)\right) \cdot \mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)\right)}^{0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) \cdot \left(2 \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \end{array} \]

Alternative 4: 60.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ t_3 := t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)\\ \mathbf{if}\;t_3 \leq 2 \cdot 10^{-319}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{n \cdot \left(2 \cdot U\right)} \cdot \sqrt[3]{t}\right) \cdot 1.5}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - t_1 \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0)))
        (t_2 (* (* 2.0 n) U))
        (t_3 (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U))))))
   (if (<= t_3 2e-319)
     (exp (* (log (* (cbrt (* n (* 2.0 U))) (cbrt t))) 1.5))
     (if (<= t_3 INFINITY)
       (sqrt (* t_2 (- (- t (* 2.0 (* l (/ l Om)))) (* t_1 (- U U*)))))
       (sqrt (* 2.0 (fabs (* U (* n (+ t (* (/ (pow l 2.0) Om) -2.0)))))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)));
	double tmp;
	if (t_3 <= 2e-319) {
		tmp = exp((log((cbrt((n * (2.0 * U))) * cbrt(t))) * 1.5));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) - (t_1 * (U - U_42_)))));
	} else {
		tmp = sqrt((2.0 * fabs((U * (n * (t + ((pow(l, 2.0) / Om) * -2.0)))))));
	}
	return tmp;
}

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * Math.pow((l / Om), 2.0);
	double t_2 = (2.0 * n) * U;
	double t_3 = t_2 * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U)));
	double tmp;
	if (t_3 <= 2e-319) {
		tmp = Math.exp((Math.log((Math.cbrt((n * (2.0 * U))) * Math.cbrt(t))) * 1.5));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) - (t_1 * (U - U_42_)))));
	} else {
		tmp = Math.sqrt((2.0 * Math.abs((U * (n * (t + ((Math.pow(l, 2.0) / Om) * -2.0)))))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = Float64(Float64(2.0 * n) * U)
	t_3 = Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U))))
	tmp = 0.0
	if (t_3 <= 2e-319)
		tmp = exp(Float64(log(Float64(cbrt(Float64(n * Float64(2.0 * U))) * cbrt(t))) * 1.5));
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) - Float64(t_1 * Float64(U - U_42_)))));
	else
		tmp = sqrt(Float64(2.0 * abs(Float64(U * Float64(n * Float64(t + Float64(Float64((l ^ 2.0) / Om) * -2.0)))))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 2e-319], N[Exp[N[(N[Log[N[(N[Power[N[(n * N[(2.0 * U), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.5), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[Abs[N[(U * N[(n * N[(t + N[(N[(N[Power[l, 2.0], $MachinePrecision] / Om), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < 1.99998e-319

   1. Initial program 8.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 11.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 8.1%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]

   5. Taylor expanded in t around inf 23.6%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 11.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. *-commutative 11.0%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

      3. associate-*l* 22.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   7. Simplified 22.0%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. sqrt-prod 21.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{n \cdot \left(U \cdot t\right)}} \]

      2. associate-*r* 11.0%

         \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{\left(n \cdot U\right) \cdot t}} \]

      3. sqrt-prod 11.0%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}} \]

      4. add-cube-cbrt 10.9%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)} \cdot \sqrt[3]{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\right) \cdot \sqrt[3]{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}}} \]

      5. unpow3 10.9%

         \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\right)}^{3}}} \]

      6. sqrt-pow1 11.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\right)}^{\left(\frac{3}{2}\right)}} \]

      7. pow-to-exp 10.6%

         \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}\right) \cdot \frac{3}{2}}} \]

      8. associate-*r* 20.6%

         \[\leadsto e^{\log \left(\sqrt[3]{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}}\right) \cdot \frac{3}{2}} \]

      9. associate-*r* 10.6%

         \[\leadsto e^{\log \left(\sqrt[3]{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}}\right) \cdot \frac{3}{2}} \]

      10. *-commutative 10.6%

          \[\leadsto e^{\log \left(\sqrt[3]{2 \cdot \color{blue}{\left(t \cdot \left(n \cdot U\right)\right)}}\right) \cdot \frac{3}{2}} \]

      11. metadata-eval 10.6%

          \[\leadsto e^{\log \left(\sqrt[3]{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\right) \cdot \color{blue}{1.5}} \]

   9. Applied egg-rr 10.6%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\right) \cdot 1.5}} \]
   10. Step-by-step derivation

       1. *-commutative 10.6%

          \[\leadsto e^{\log \left(\sqrt[3]{\color{blue}{\left(t \cdot \left(n \cdot U\right)\right) \cdot 2}}\right) \cdot 1.5} \]

       2. *-commutative 10.6%

          \[\leadsto e^{\log \left(\sqrt[3]{\color{blue}{\left(\left(n \cdot U\right) \cdot t\right)} \cdot 2}\right) \cdot 1.5} \]

       3. *-commutative 10.6%

          \[\leadsto e^{\log \left(\sqrt[3]{\color{blue}{2 \cdot \left(\left(n \cdot U\right) \cdot t\right)}}\right) \cdot 1.5} \]

       4. associate-*r* 10.7%

          \[\leadsto e^{\log \left(\sqrt[3]{\color{blue}{\left(2 \cdot \left(n \cdot U\right)\right) \cdot t}}\right) \cdot 1.5} \]

       5. associate-*l* 10.7%

          \[\leadsto e^{\log \left(\sqrt[3]{\color{blue}{\left(\left(2 \cdot n\right) \cdot U\right)} \cdot t}\right) \cdot 1.5} \]

       6. cbrt-prod 36.1%

          \[\leadsto e^{\log \color{blue}{\left(\sqrt[3]{\left(2 \cdot n\right) \cdot U} \cdot \sqrt[3]{t}\right)} \cdot 1.5} \]

       7. associate-*l* 36.0%

          \[\leadsto e^{\log \left(\sqrt[3]{\color{blue}{2 \cdot \left(n \cdot U\right)}} \cdot \sqrt[3]{t}\right) \cdot 1.5} \]

   11. Applied egg-rr 36.0%

       \[\leadsto e^{\log \color{blue}{\left(\sqrt[3]{2 \cdot \left(n \cdot U\right)} \cdot \sqrt[3]{t}\right)} \cdot 1.5} \]
   12. Step-by-step derivation

       1. *-commutative 36.0%

          \[\leadsto e^{\log \left(\sqrt[3]{2 \cdot \color{blue}{\left(U \cdot n\right)}} \cdot \sqrt[3]{t}\right) \cdot 1.5} \]

       2. associate-*r* 36.1%

          \[\leadsto e^{\log \left(\sqrt[3]{\color{blue}{\left(2 \cdot U\right) \cdot n}} \cdot \sqrt[3]{t}\right) \cdot 1.5} \]

       3. *-commutative 36.1%

          \[\leadsto e^{\log \left(\sqrt[3]{\color{blue}{n \cdot \left(2 \cdot U\right)}} \cdot \sqrt[3]{t}\right) \cdot 1.5} \]

   13. Simplified 36.1%

       \[\leadsto e^{\log \color{blue}{\left(\sqrt[3]{n \cdot \left(2 \cdot U\right)} \cdot \sqrt[3]{t}\right)} \cdot 1.5} \]

   if 1.99998e-319 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))) < +inf.0

   1. Initial program 66.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 68.6%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 68.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))

   1. Initial program 0.0%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 0.4%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 0.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)}\right)}} \]

      2. sqrt-unprod 0.4%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{\left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}}} \]

      3. pow2 0.4%

         \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{{\left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}^{2}}}} \]

   5. Applied egg-rr 0.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\sqrt{{\left(n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{2}}}} \]
   6. Step-by-step derivation

      1. unpow2 0.7%

         \[\leadsto \sqrt{2 \cdot \sqrt{\color{blue}{\left(n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right) \cdot \left(n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}}} \]

      2. rem-sqrt-square 0.7%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left|n \cdot \left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{{\ell}^{2}}{Om}, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right)\right|}} \]

      3. fma-def 0.7%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(t - \color{blue}{\left(2 \cdot \frac{{\ell}^{2}}{Om} + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)\right)\right|} \]

      4. associate--r+ 0.7%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \color{blue}{\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)}\right)\right|} \]

      5. cancel-sign-sub-inv 0.7%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\color{blue}{\left(t + \left(-2\right) \cdot \frac{{\ell}^{2}}{Om}\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      6. metadata-eval 0.7%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\left(t + \color{blue}{-2} \cdot \frac{{\ell}^{2}}{Om}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      7. +-commutative 0.7%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      8. *-commutative 0.7%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\left(\color{blue}{\frac{{\ell}^{2}}{Om} \cdot -2} + t\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      9. fma-def 0.7%

         \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right)} - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)\right|} \]

      10. associate-*r* 0.7%

          \[\leadsto \sqrt{2 \cdot \left|n \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\right)\right)\right|} \]

   7. Simplified 0.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left|n \cdot \left(U \cdot \left(\mathsf{fma}\left(\frac{{\ell}^{2}}{Om}, -2, t\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)\right|}} \]

   8. Taylor expanded in n around 0 44.4%

      \[\leadsto \sqrt{2 \cdot \left|\color{blue}{U \cdot \left(n \cdot \left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)}\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq 2 \cdot 10^{-319}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{n \cdot \left(2 \cdot U\right)} \cdot \sqrt[3]{t}\right) \cdot 1.5}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right) \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left|U \cdot \left(n \cdot \left(t + \frac{{\ell}^{2}}{Om} \cdot -2\right)\right)\right|}\\ \end{array} \]

Alternative 5: 55.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\\ t_2 := \left(2 \cdot n\right) \cdot U\\ \mathbf{if}\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + t_1 \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - t_1 \cdot \left(U - U*\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* n (pow (/ l Om) 2.0))) (t_2 (* (* 2.0 n) U)))
   (if (<=
        (sqrt (* t_2 (+ (- t (* 2.0 (/ (* l l) Om))) (* t_1 (- U* U)))))
        0.0)
     (* (sqrt (* 2.0 n)) (sqrt (* U t)))
     (sqrt (* t_2 (- (- t (* 2.0 (* l (/ l Om)))) (* t_1 (- U U*))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * pow((l / Om), 2.0);
	double t_2 = (2.0 * n) * U;
	double tmp;
	if (sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U))))) <= 0.0) {
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	} else {
		tmp = sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) - (t_1 * (U - U_42_)))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = n * ((l / om) ** 2.0d0)
    t_2 = (2.0d0 * n) * u
    if (sqrt((t_2 * ((t - (2.0d0 * ((l * l) / om))) + (t_1 * (u_42 - u))))) <= 0.0d0) then
        tmp = sqrt((2.0d0 * n)) * sqrt((u * t))
    else
        tmp = sqrt((t_2 * ((t - (2.0d0 * (l * (l / om)))) - (t_1 * (u - u_42)))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = n * Math.pow((l / Om), 2.0);
	double t_2 = (2.0 * n) * U;
	double tmp;
	if (Math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U))))) <= 0.0) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((U * t));
	} else {
		tmp = Math.sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) - (t_1 * (U - U_42_)))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = n * math.pow((l / Om), 2.0)
	t_2 = (2.0 * n) * U
	tmp = 0
	if math.sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U))))) <= 0.0:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((U * t))
	else:
		tmp = math.sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) - (t_1 * (U - U_42_)))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(n * (Float64(l / Om) ^ 2.0))
	t_2 = Float64(Float64(2.0 * n) * U)
	tmp = 0.0
	if (sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) + Float64(t_1 * Float64(U_42_ - U))))) <= 0.0)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(U * t)));
	else
		tmp = sqrt(Float64(t_2 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) - Float64(t_1 * Float64(U - U_42_)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = n * ((l / Om) ^ 2.0);
	t_2 = (2.0 * n) * U;
	tmp = 0.0;
	if (sqrt((t_2 * ((t - (2.0 * ((l * l) / Om))) + (t_1 * (U_42_ - U))))) <= 0.0)
		tmp = sqrt((2.0 * n)) * sqrt((U * t));
	else
		tmp = sqrt((t_2 * ((t - (2.0 * (l * (l / Om)))) - (t_1 * (U - U_42_)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(t$95$2 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*))))) < 0.0

   1. Initial program 7.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 7.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 7.2%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]

   5. Taylor expanded in t around inf 23.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 7.2%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. *-commutative 7.2%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

      3. associate-*l* 23.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   7. Simplified 23.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 23.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot t\right)}} \]

      2. sqrt-prod 34.8%

         \[\leadsto \color{blue}{\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}} \]

      3. *-commutative 34.8%

         \[\leadsto \sqrt{\color{blue}{n \cdot 2}} \cdot \sqrt{U \cdot t} \]

   9. Applied egg-rr 34.8%

      \[\leadsto \color{blue}{\sqrt{n \cdot 2} \cdot \sqrt{U \cdot t}} \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 2 n) U) (-.f64 (-.f64 t (*.f64 2 (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) 2)) (-.f64 U U*)))))

   1. Initial program 56.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 58.9%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 58.9%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) + \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right)\right)} \leq 0:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array} \]

Alternative 6: 53.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -0.94 \lor \neg \left(U \leq 3.1 \cdot 10^{+80}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (or (<= U -0.94) (not (<= U 3.1e+80)))
   (sqrt (* 2.0 (* (- t (* 2.0 (* l (/ l Om)))) (* n U))))
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (+
       (+ t (* -2.0 (/ l (/ Om l))))
       (* n (* (pow (/ l Om) 2.0) (- U* U)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U <= -0.94) || !(U <= 3.1e+80)) {
		tmp = sqrt((2.0 * ((t - (2.0 * (l * (l / Om)))) * (n * U))));
	} else {
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (pow((l / Om), 2.0) * (U_42_ - U)))))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if ((u <= (-0.94d0)) .or. (.not. (u <= 3.1d+80))) then
        tmp = sqrt((2.0d0 * ((t - (2.0d0 * (l * (l / om)))) * (n * u))))
    else
        tmp = sqrt(((2.0d0 * n) * (u * ((t + ((-2.0d0) * (l / (om / l)))) + (n * (((l / om) ** 2.0d0) * (u_42 - u)))))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if ((U <= -0.94) || !(U <= 3.1e+80)) {
		tmp = Math.sqrt((2.0 * ((t - (2.0 * (l * (l / Om)))) * (n * U))));
	} else {
		tmp = Math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (Math.pow((l / Om), 2.0) * (U_42_ - U)))))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if (U <= -0.94) or not (U <= 3.1e+80):
		tmp = math.sqrt((2.0 * ((t - (2.0 * (l * (l / Om)))) * (n * U))))
	else:
		tmp = math.sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (math.pow((l / Om), 2.0) * (U_42_ - U)))))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if ((U <= -0.94) || !(U <= 3.1e+80))
		tmp = sqrt(Float64(2.0 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) * Float64(n * U))));
	else
		tmp = sqrt(Float64(Float64(2.0 * n) * Float64(U * Float64(Float64(t + Float64(-2.0 * Float64(l / Float64(Om / l)))) + Float64(n * Float64((Float64(l / Om) ^ 2.0) * Float64(U_42_ - U)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if ((U <= -0.94) || ~((U <= 3.1e+80)))
		tmp = sqrt((2.0 * ((t - (2.0 * (l * (l / Om)))) * (n * U))));
	else
		tmp = sqrt(((2.0 * n) * (U * ((t + (-2.0 * (l / (Om / l)))) + (n * (((l / Om) ^ 2.0) * (U_42_ - U)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[Or[LessEqual[U, -0.94], N[Not[LessEqual[U, 3.1e+80]], $MachinePrecision]], N[Sqrt[N[(2.0 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * N[(U * N[(N[(t + N[(-2.0 * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(n * N[(N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(U$42$ - U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < -0.93999999999999995 or 3.09999999999999988e80 < U

   1. Initial program 51.2%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 49.5%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]
   5. Step-by-step derivation

      1. unpow2 49.5%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

      2. associate-*l/ 53.0%

         \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right)\right)} \]

   6. Applied egg-rr 53.0%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right)\right)} \]

   if -0.93999999999999995 < U < 3.09999999999999988e80

   1. Initial program 50.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 54.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -0.94 \lor \neg \left(U \leq 3.1 \cdot 10^{+80}\right):\\ \;\;\;\;\sqrt{2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t + -2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right)\right)\right)}\\ \end{array} \]

Alternative 7: 46.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt (* 2.0 (* (- t (* 2.0 (* l (/ l Om)))) (* n U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * ((t - (2.0 * (l * (l / Om)))) * (n * U))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * ((t - (2.0d0 * (l * (l / om)))) * (n * u))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * ((t - (2.0 * (l * (l / Om)))) * (n * U))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * ((t - (2.0 * (l * (l / Om)))) * (n * U))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(Float64(t - Float64(2.0 * Float64(l * Float64(l / Om)))) * Float64(n * U))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * ((t - (2.0 * (l * (l / Om)))) * (n * U))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(N[(t - N[(2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 49.4%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in Om around inf 45.2%

   \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 45.2%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{Om}\right)\right)} \]

   2. associate-*l/ 46.8%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right)\right)} \]

6. Applied egg-rr 46.8%

   \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right)\right)} \]

7. Final simplification 46.8%

   \[\leadsto \sqrt{2 \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(n \cdot U\right)\right)} \]

Alternative 8: 35.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+158}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t -2.5e+158)
   (pow (* 2.0 (* n (* U t))) 0.5)
   (if (<= t 3.7e+123)
     (sqrt (* 2.0 (* t (* n U))))
     (sqrt (* 2.0 (* U (* n t)))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -2.5e+158) {
		tmp = pow((2.0 * (n * (U * t))), 0.5);
	} else if (t <= 3.7e+123) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= (-2.5d+158)) then
        tmp = (2.0d0 * (n * (u * t))) ** 0.5d0
    else if (t <= 3.7d+123) then
        tmp = sqrt((2.0d0 * (t * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= -2.5e+158) {
		tmp = Math.pow((2.0 * (n * (U * t))), 0.5);
	} else if (t <= 3.7e+123) {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= -2.5e+158:
		tmp = math.pow((2.0 * (n * (U * t))), 0.5)
	elif t <= 3.7e+123:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= -2.5e+158)
		tmp = Float64(2.0 * Float64(n * Float64(U * t))) ^ 0.5;
	elseif (t <= 3.7e+123)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= -2.5e+158)
		tmp = (2.0 * (n * (U * t))) ^ 0.5;
	elseif (t <= 3.7e+123)
		tmp = sqrt((2.0 * (t * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, -2.5e+158], N[Power[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[t, 3.7e+123], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.4999999999999998e158

   1. Initial program 42.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 42.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 34.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 47.1%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

      2. pow-to-exp 32.2%

         \[\leadsto \color{blue}{e^{\log \left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right) \cdot 0.5}} \]

      3. associate-*r* 28.7%

         \[\leadsto e^{\log \left(2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}\right) \cdot 0.5} \]

      4. *-commutative 28.7%

         \[\leadsto e^{\log \left(2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)\right) \cdot 0.5} \]

   6. Applied egg-rr 28.7%

      \[\leadsto \color{blue}{e^{\log \left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right) \cdot 0.5}} \]
   7. Step-by-step derivation

      1. exp-to-pow 43.0%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}^{0.5}} \]

      2. associate-*l* 47.6%

         \[\leadsto {\left(2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}\right)}^{0.5} \]

   8. Applied egg-rr 47.6%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}} \]

   if -2.4999999999999998e158 < t < 3.69999999999999996e123

   1. Initial program 49.1%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 47.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 26.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 31.9%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. *-commutative 31.9%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   6. Simplified 31.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]

   if 3.69999999999999996e123 < t

   1. Initial program 61.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 62.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 61.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+158}:\\ \;\;\;\;{\left(2 \cdot \left(n \cdot \left(U \cdot t\right)\right)\right)}^{0.5}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 9: 36.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.75 \cdot 10^{+112}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.75e+112)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.75e+112) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 1.75d+112) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.75e+112) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 1.75e+112:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.75e+112)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 1.75e+112)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.75e+112], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.74999999999999998e112

   1. Initial program 48.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 46.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 27.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 29.4%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 33.8%

         \[\leadsto {\left(2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}\right)}^{0.5} \]

      3. *-commutative 33.8%

         \[\leadsto {\left(2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)\right)}^{0.5} \]

   6. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}^{0.5}} \]

   if 1.74999999999999998e112 < t

   1. Initial program 61.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 62.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 61.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.75 \cdot 10^{+112}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 10: 37.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+123}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 6e+123)
   (pow (* 2.0 (* t (* n U))) 0.5)
   (pow (* (* 2.0 U) (* n t)) 0.5)))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 6e+123) {
		tmp = pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = pow(((2.0 * U) * (n * t)), 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 6d+123) then
        tmp = (2.0d0 * (t * (n * u))) ** 0.5d0
    else
        tmp = ((2.0d0 * u) * (n * t)) ** 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 6e+123) {
		tmp = Math.pow((2.0 * (t * (n * U))), 0.5);
	} else {
		tmp = Math.pow(((2.0 * U) * (n * t)), 0.5);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 6e+123:
		tmp = math.pow((2.0 * (t * (n * U))), 0.5)
	else:
		tmp = math.pow(((2.0 * U) * (n * t)), 0.5)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 6e+123)
		tmp = Float64(2.0 * Float64(t * Float64(n * U))) ^ 0.5;
	else
		tmp = Float64(Float64(2.0 * U) * Float64(n * t)) ^ 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 6e+123)
		tmp = (2.0 * (t * (n * U))) ^ 0.5;
	else
		tmp = ((2.0 * U) * (n * t)) ^ 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 6e+123], N[Power[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], N[Power[N[(N[(2.0 * U), $MachinePrecision] * N[(n * t), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.00000000000000016e123

   1. Initial program 48.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 46.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 27.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. pow1/2 29.4%

         \[\leadsto \color{blue}{{\left(2 \cdot \left(U \cdot \left(n \cdot t\right)\right)\right)}^{0.5}} \]

      2. associate-*r* 33.8%

         \[\leadsto {\left(2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}\right)}^{0.5} \]

      3. *-commutative 33.8%

         \[\leadsto {\left(2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)\right)}^{0.5} \]

   6. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(\left(n \cdot U\right) \cdot t\right)\right)}^{0.5}} \]

   if 6.00000000000000016e123 < t

   1. Initial program 61.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 62.0%

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied egg-rr 62.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot \ell\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   4. Taylor expanded in t around inf 61.8%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 61.8%

         \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]

   6. Simplified 61.8%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot U\right) \cdot \left(n \cdot t\right)}} \]
   7. Step-by-step derivation

      1. pow1/2 64.0%

         \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}} \]

   8. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6 \cdot 10^{+123}:\\ \;\;\;\;{\left(2 \cdot \left(t \cdot \left(n \cdot U\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(2 \cdot U\right) \cdot \left(n \cdot t\right)\right)}^{0.5}\\ \end{array} \]

Alternative 11: 35.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 5e-176) (sqrt (* 2.0 (* n (* U t)))) (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 5e-176) {
		tmp = sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 5d-176) then
        tmp = sqrt((2.0d0 * (n * (u * t))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 5e-176) {
		tmp = Math.sqrt((2.0 * (n * (U * t))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 5e-176:
		tmp = math.sqrt((2.0 * (n * (U * t))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 5e-176)
		tmp = sqrt(Float64(2.0 * Float64(n * Float64(U * t))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 5e-176)
		tmp = sqrt((2.0 * (n * (U * t))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 5e-176], N[Sqrt[N[(2.0 * N[(n * N[(U * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5e-176

   1. Initial program 46.6%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 47.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in Om around inf 41.0%

      \[\leadsto \sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \color{blue}{2 \cdot \frac{{\ell}^{2}}{Om}}\right)\right)} \]

   5. Taylor expanded in t around inf 24.7%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*r* 30.0%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. *-commutative 30.0%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

      3. associate-*l* 27.5%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   7. Simplified 27.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(n \cdot \left(U \cdot t\right)\right)}} \]

   if 5e-176 < t

   1. Initial program 56.7%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 47.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 12: 35.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 4.4e+120)
   (sqrt (* 2.0 (* t (* n U))))
   (sqrt (* 2.0 (* U (* n t))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 4.4e+120) {
		tmp = sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 4.4d+120) then
        tmp = sqrt((2.0d0 * (t * (n * u))))
    else
        tmp = sqrt((2.0d0 * (u * (n * t))))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 4.4e+120) {
		tmp = Math.sqrt((2.0 * (t * (n * U))));
	} else {
		tmp = Math.sqrt((2.0 * (U * (n * t))));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 4.4e+120:
		tmp = math.sqrt((2.0 * (t * (n * U))))
	else:
		tmp = math.sqrt((2.0 * (U * (n * t))))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 4.4e+120)
		tmp = sqrt(Float64(2.0 * Float64(t * Float64(n * U))));
	else
		tmp = sqrt(Float64(2.0 * Float64(U * Float64(n * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 4.4e+120)
		tmp = sqrt((2.0 * (t * (n * U))));
	else
		tmp = sqrt((2.0 * (U * (n * t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 4.4e+120], N[Sqrt[N[(2.0 * N[(t * N[(n * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.4000000000000003e120

   1. Initial program 48.3%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 46.7%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 27.9%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 31.8%

         \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(U \cdot n\right) \cdot t\right)}} \]

      2. *-commutative 31.8%

         \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\left(n \cdot U\right)} \cdot t\right)} \]

   6. Simplified 31.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(\left(n \cdot U\right) \cdot t\right)}} \]

   if 4.4000000000000003e120 < t

   1. Initial program 61.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Simplified 62.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf 61.8%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.4 \cdot 10^{+120}:\\ \;\;\;\;\sqrt{2 \cdot \left(t \cdot \left(n \cdot U\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}\\ \end{array} \]

Alternative 13: 35.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* 2.0 (* U (* n t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((2.0 * (U * (n * t))));
}

Fortran

real(8) function code(n, u, t, l, om, u_42)
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((2.0d0 * (u * (n * t))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((2.0 * (U * (n * t))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((2.0 * (U * (n * t))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(2.0 * Float64(U * Float64(n * t))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((2.0 * (U * (n * t))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(2.0 * N[(U * N[(n * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.6%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Simplified 49.4%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(n \cdot \left(U - U*\right)\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf 33.8%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\left(U \cdot \left(n \cdot t\right)\right)}} \]

5. Final simplification 33.8%

   \[\leadsto \sqrt{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)} \]

Reproduce

herbie shell --seed 2023301 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))